JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE) ›› 2024, Vol. 59 ›› Issue (12): 73-78.doi: 10.6040/j.issn.1671-9352.0.2023.193

Previous Articles     Next Articles

Julia limiting directions of entire solutions of complex differential equations

XIANG Xuxu, LIU Jianming, WANG Qin, OUYANG Ruiqi   

  1. School of Mathematical Sciences, Guizhou Normal University, Guiyang 550025, Guizhou, China
  • Published:2024-12-12

PDF (PC)

17

Abstract: The Julia limiting directions of entire solutions of complex differential equations f (n)+Pn-1(z)f (n-1)+… +P0(z)f=0 and f ″+A(z)f '+B(z)f=0 are studied by using Phragmén-Lindelöf indicator function and the properties of the completely regular growth functions, the lower bound of the measure of the set of common Julia limiting directions of the derivatives and primitives of infinite order entire solutions of the equations mentioned above is obtained, where n≥2, Pj(z)(j=1,2,…,n-1) are entire functions, P0(z), A(z) and B(z) are entire functions of completely regular growth.

Key words: complex differential equation, the limiting direction of Julia set, Phragmén-Lindelö, f indicator function, completely regular growth

CLC Number: 

  • O174.52
[1] LAINE I. Nevanlinna theory and complex differential equations[M]. Berlin: Walter de Gruyter, 1993.
[2] HAYMAN W K. Meromorphic functions[M]. Oxford: Clarendon Press, 1964.
[3] 杨乐. 值分布论及其新研究[M]. 北京:科学出版社,1982. YANG Le. Value distribution theory and its new research[M]. Beijing: Science Press, 1982.
[4] YANG Chungchun, YI Hongxun. Uniqueness theory of meromorphic functions[M]. Dordrech: Kluwer Academic Publishers Group, 2003.
[5] 郑建华. 亚纯函数动力系统[M]. 北京:清华大学出版社,2006. ZHENG Jianhua. Dynamic system of meromorphic functions[M]. Beijing: Tsinghua University Press, 2006.
[6] BAKER I N. Sets of non-normality in iteration theory[J]. Journal of London Mathematical Society, 1965, 40:499-502.
[7] QIAO Jianyong. On limiting directions of Julia sets[J]. Annales Academiae Scientiarum Fennicae: Mathmatica, 2001, 26(2):391-399.
[8] ZHENG Jianhua, WANG Sheng, HUANG Zhigang. Some properties of Fatou and Julia sets of transcendental meromorphic functions[J]. Bulletin of the Australian Mathematical Society, 2002, 66(1):1-8.
[9] HUANG Zhigang, WANG Jun. On the radial distribution of Julia sets of entire solutions of f (n)+A(z)f=0[J]. Journal of Mathematical Analysis and Applications, 2012, 387(2):1106-1113.
[10] 龙见仁,吴兴群,陈寒霜.整函数系数复微分方程解的快速增长性[J].贵州师范大学学报(自然科学版),2021, 39(3):8-14. LONG Jianren,WU Xingqun,CHEN Hanshuang.Fast growth of solutions of complex differential equations with entire coefficients[J]. Journal of Guizhou Normal University(Natural Sciences), 2021, 39(3):8-14.
[11] 王钦,龙见仁. 有穷φ级亚纯函数与线性Jackson q-差分方程[J]. 贵州师范大学学报(自然科学版),2024, 42(4):123-128. WANG Qin, LONG Jianren. Finite φ-order of meromorphic functions and linear Jackson q-divided difference equations[J].Journal of Guizhou Normal University(Natural Sciences), 2024, 42(4):123-128.
[12] WANG Jun, YAO Xiao. On Julia limiting directions of meromorphic functions[J]. Israel Journal of Mathematics, 2020, 238(1):405-430.
[13] WANG Jun, YAO Xiao, ZHANG Chengchun. Julia limiting directions of entire solutions of complex differential equations[J]. Acta Mathematica Scientia, 2021, 41(4):1275-1286.
[14] 张国威,丁杰,杨连中. 复线性微分方程解的导数的Julia集的径向分布[J]. 中国科学:数学,2014,44(6):693-700. ZHANG Guowei, DING Jie, YANG Lianzhong. Radial distribution of Julia sets of derivatives of solutions to complex linear differential equations[J]. Scientia Sinica Mathematica, 2014, 44(6):693-700.
[15] WANG Jun, CHEN Zongxuan. Limiting directions of Julia sets of entire solutions to complex differential equations[J]. Acta Mathematica Scientia, 2017, 37(1):97-107.
[16] LEVIN B I. Distribution of zeros of entire functions[M]. Providence: American Mathematical Society, 1964.
[17] RONKIN L I. Functions of completely regular growth[M]. Dordrecht: Kluwer Academic Publishers Group, 1992.
[18] HUANG Zhibo, LUO Minwei, CHEN Zongxuan. The growth of solutions to higher order differential equations with exponential polynomials as its coefficients[J]. Acta Mathematica Scientia, 2023, 43(1):439-449.
[19] GOLDBERG A A, OSTROVSKII I V. Value distribution of meromorphic function[M] //Translations of Mathematical Monographs Series. Providence: Amer Math Soc, 2008: 488.
[20] ZHENG Jianhua. Value distribution of meromorphic functions[M]. Beijing: Springer Science & Business Media, 2011.
[21] HUANG Zhigang, WANG Jun. On the radial distribution of Julia sets of entire solutions of f (n)+A(z)f=0[J]. Journal of Mathematical Analysis and Applications, 2012, 387(2):1106-1113.
[22] BAKER I N. The domains of normality of an entire function[J]. Annales Academiae Scientiarum Fennicae, 1975(1):277-283.
[23] 高仕安,陈宗煊,陈特为. 线性微分方程的复振荡理论[M]. 武汉:华中理工大学出版社, 1998. GAO Shian, CHEN Zongxuan, CHEN Tewei. Theory of complex oscillation of linear differential equation[M]. Wuhan: Huazhong University of Science and Technology Press, 1998.
[1] LI Yuan-fei, LI Dan-dan, CHEN Xue-jiao, SHI Jin-cheng. Alternative results of a class of quasilinear transient parabolic equations [J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2021, 56(6): 1-9.
[2] LI Yuan-fei， CHEN Xue-jiao， SHI Jin-cheng. Phragmén-Lindelof alternative for the heat conduction equations in a binary mixture [J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2020, 55(12): 1-12.
[3] ZHANG Guo-wei. Radial distribution of Julia sets of derivatives of solutions to complex differential equations [J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2016, 51(2): 85-88.
[4] SU Xian-feng, LI Xiao-meng. On non-admissible meromorphic solutions of systems of  non-linear complex algebraic differential equations [J]. J4, 2012, 47(8): 39-41.
[5] XU Jun-feng. On the growth of the meromorphic solutions of complex algebraic differential equations [J]. J4, 2010, 45(6): 91-93.
Viewed
Full text


Abstract

Cited
  Shared   
  Discussed   
[1] LI Bin1,2, LI Yi-bin1, RONG Xue-wen1. Intelligent optimization strategy for ELM-RBF neural networks[J]. J4, 2010, 45(5): 48 -51 .
[2] DAI Tao1, XI Kai-hua2*, DAI Jia-lin1, LU Tong-chao2, YU Jin-biao1, REN Yong-qiang2, YANG Yao-zhong1, CHENG Ai-jie2. A method for numerical simulation of binary combination flooding based on interpolation of capillary number[J]. J4, 2012, 47(8): 55 -59 .
[3] TANG Qian, YANG Fei, HUANG Qi, LIN Guo-yuan. Security transfer model of access control information based on TCB subsets[J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2016, 51(7): 98 -106 .
[4] ZHANG Dong-qing, YIN Xiao-bin, GAO Han-peng. Quasi-linearly Armendariz modules[J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2016, 51(12): 1 -6 .
[5] ZHOU Jian-wei1, YANG Dan-ping2*. A posteriori error estimates for Legendre-Galerkin spectral  methods in two dimensions[J]. J4, 2011, 46(11): 122 -126 .
[6] HUANG Lei-lei, SONG Xiao-qiu, LU Wei. On polynomial stability of linear discrete-time systems in Banach spaces[J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2017, 52(10): 36 -41 .
[7] WEN Chang-kun, REN Hai-zhen. On the Wiener index of triangular chains[J]. J4, 2013, 48(2): 53 -56 .
[8] LIU Fu-ti1,2, CHENG Xiao-hong2, ZHANG Shu-hua3. First principles study of structure and thermodynamic properties of MgH2[J]. J4, 2012, 47(1): 39 -43 .
[9] TAN Cheng, ZHANG Huan-shui. Lyapunov-type stabilizating conditions of discrete-time stochastic systems with input delay[J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2016, 51(5): 114 -120 .
[10] LI Jia-qi, DENG Yu-zhou, LIU Gang, YUE Ren-liang, YANG Jun, CHEN Yun-fa. The application of flame combustion synthesis in the preparation of metal oxide and supported noble metal catalysts[J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2016, 51(1): 1 -13 .
Copyright 2018 © Editorial office of JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE)
Supported by Beijing Magtech Co.Ltd